In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks. To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matrices associated with edges in this set have a non-trivial intersection of null spaces. We show that necessary and/or sufficient conditions for bipartite consensus on matrix-weighted networks can be characterized by the uniqueness of the non-trivial balancing set, while the contribution of the associated non-trivial intersection of null spaces to the steady-state of the matrix-weighted network is examined. Moreover, for matrix-weighted networks with a positive-negative spanning tree, necessary and sufficient condition for bipartite consensus using the non-trivial balancing set is obtained. Simulation examples are provided to demonstrate the theoretical results.
Laplacian dynamics on signed digraphs have a richer behavior than those on nonnegative digraphs. In particular, for the so-called "repelling" signed Laplacians, the marginal stability property (needed to achieve consensus) is not guaranteed a priori and, even when it holds, it does not automatically lead to consensus, as these signed Laplacians may loose rank even in strongly connected digraphs. Furthermore, in the time-varying case, instability can occur even when switching in a family of systems each of which corresponds to a marginally stable signed Laplacian with the correct corank. In thi…
Grounded Laplacians of Directed Signed Matrix-Weighted Networks: Spectral Properties and Applications to Non-Trivial Consensus
the College of Computer Science and Software Engineering, Shenzhen University（深圳大学计算机科学与软件工程学院）；the School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；the Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology（深圳先进技术大学计算机与控制工程学院）
In this work, we present new secondary regulators for current sharing and voltage balancing in DC microgrids, composed of distributed generation units, dynamic RLC lines, and nonlinear ZIP (constant impedance, constant current, and constant power) loads. The proposed controllers sit atop a primary voltage control layer, and exchange information over a communication network to perform secondary control actions. We deduce sufficient conditions for the existence and uniqueness of an equilibrium point, and show that the desired objectives are attained in steady state. Our control design requires o…
In this work, we present a novel consensus-based secondary control scheme for current sharing and voltage balancing in DC microgrids, composed of distributed generation units, dynamic RLC lines, and nonlinear ZIE (constant impedance, constant current, and exponential) loads. Situated atop a primary voltage control layer, our secondary controllers have a distributed structure, and utilize information exchanged over a communication network to compute necessary control actions. Besides showing that the desired objectives are always attained in steady state, we deduce sufficient conditions for the…
Non-trivial consensus on directed signed matrix-weighted networks with compound measurement noises and time-varying topologies
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；College of Computer Science and Software Engineering, Shenzhen University（深圳大学计算机科学与软件工程学院）
Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities
Delft University of Technology（代尔夫特理工大学）；ITMO University（ITMO大学）；Institute for Problems of Mechanical Engineering of Russian Academy of Sciences (IPME RAS)（俄罗斯科学院机械问题研究所）；University of Groningen（格罗宁根大学）
This paper examines the stability and distributed stabilization of signed multi-agent networks. Here, positive semidefiniteness is not inherent for signed Laplacians, which renders the stability and consensus of this category of networks intricate. First, we examine the stability of signed networks by introducing a novel graph-theoretic objective negative cut set, which implies that manipulating negative edge weights cannot change a unstable network into a stable one. Then, inspired by the diagonal dominance and stability of matrices, a local state damping mechanism is introduced using self-lo…
In this paper, we address two forms of consensus for multi-agent systems with undirected, signed, weighted, and connected communication graphs, under the assumption that the agents can be partitioned into three clusters, representing the decision classes on a given specific topic, for instance, the in favour, abstained and opponent agents. We will show that under some assumptions on the cooperative/antagonistic relationships among the agents, simple modifications of DeGroot's algorithm allow to achieve tripartite consensus(if the opinions of agents belonging to the same class all converge to t…
Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization
University of Groningen（格罗宁根大学）；St. Petersburg State University（圣彼得堡国立大学）；ITMO University（ITMO大学）；Institute for Problems of Mechanical Engineering of Russian Academy of Sciences (IPME RAS)（俄罗斯科学院机械问题研究所）
Trust and distrust are common in the opinion interactions among agents in social networks, and they are described by the edges with positive and negative weights in the signed digraph, respectively. It has been shown in social psychology that although the opinions of most agents (followers) tend to prevail, sometimes one agent (leader) with a firm stand and strong influence can impact or even overthrow the preferences of followers. This paper aims to analyze how the leader influences the formation of followers' opinions in signed social networks. In addition, this paper considers an asynchrono…
Attitudinal Network Graphs are signed graphs where edges capture an expressed opinion; two vertices connected by an edge can be agreeable (positive) or antagonistic (negative). A signed graph is called balanced if each of its cycles includes an even number of negative edges. Balance is often characterized by the frustration index or by finding a single convergent balanced state of network consensus. In this paper, we propose to expand the measures of consensus from a single balanced state associated with the frustration index to the set of nearest balanced states. We introduce the frustration …
Achieving consensus via nearest neighbor rules is an important prerequisite for multi-agent networks to accomplish collective tasks. A common assumption in consensus setup is that each agent interacts with all its neighbors. This paper examines whether network functionality and performance can be maintained-and even enhanced-when agents interact only with a subset of their respective (available) neighbors. As shown in the paper, the answer to this inquiry is affirmative. In this direction, we show that by exploring the monotonicity property of the Laplacian eigenvectors, a neighbor selection r…
